Langlands correspondence and Bezrukavnikov's equivalence
Course given at the University of Sydney, first semester 2019.

A course in two parts:
1) an attempt to explain what the Langlands program is about from an arithmetical point of view;
2) affine Hecke algebras, Deligne-Langlands conjecture and Bezrukavnikov's equivalence.

Extended abstract and bibliography

Notes by Anna Romanova

(Anna's notes also contain a more complete bibliography for part 1 of the course.)

Part 1

Below are hand-written notes and some handouts for the course, produced as it progressed.

Notes from Gus Lehrer's course last semester on algebraic number theory.

Lecture 1: Introduction to reciprocity

    Number of solutions handout for first lecture.

Lecture 2: Review of algebraic number theory

  Degree 5 solutions handout for second lecture.

Lecture 3: Zeta function and L-functions

    Excerpt from Mazur-Stein giving Riemann's approximations to the zeta function via more and more roots.

Lecture 4: (Gus' lecture, notes thanks to Bregje Pauwels) Artin L-functions

Lecture 5: (Gus' second lecture, notes thanks to Bregje Pauwels) Brauer's induction theorem

Lecture 6: Overview of the Sato-Tate conjecture

    Example sheet of first 5000 primes for two curves.

    From slides of a lecture by Ito containing a manuscript of Sato.

Lecture 7: Infinite Galois theory, overview of global class field theory (Artin's point of view).

Lecture 8: Structure of local Galois groups; local class field theory.

    Two interesting historical accounts:

    K. Conrad: History of class field theory.

    K. Miyake: Takagi's Class Field Theory -- From where? and to where?

Lecture 9: Heuristic derivation of local Langlands for GL(2); basic rep theory of p-adic groups.

Lecture 10: Precise statement of local Langlands for GL(2) for p ≠ 2.

Lecture 11: Why is p = 2 special? Spherical representations and Satake isomorphism.

Lecture 12: Deligne-Langlands conjecture, affine Hecke algebras and Kazhdan-Lusztig equivalence, rough statement of Bezrukavnikov's equivalence.